A dynamical system is one whose evolution is deterministic in the sense that its future motion is determined by its current state and its past history.
A dynamical system that exhibits chaos is characterized principally by Sensitivity to Initial Conditions. A well known example of this is the weather. Small changes in weather patterns can result in large changes later on. This is one reason why, although meteorologists can fairly well predict if it will rain tomorrow, they can't determine if it will be raining a year from now.
Before chaos was recognized as a separate class of dynamics, it was thought of as randomness and indeterminacy. In fact, this notion was held steadfastly despite many observations of systems that actually evolved in an almost periodic manner for a short time before returning to a more random behavior. Not until James Yorke named this class of dynamics chaos did it start to take its distinction from randomness and indeterminacy. Following the terminology came the first widely popular definition of chaos: the superposition of a very large (infinite) number of periodic motions. This means that a chaotic system may spend a short time in an almost periodic motion, then it may transition to another periodic motion with period that is, say, 20 times the previous one. Such evolution from one unstable periodic motion to the next would give an overall impression of randomness, but in a short term analysis it would show order. This observation was not misleading since chaos is deterministic and not random in nature.
A second definition of chaos is that the resultant system dynamics are sensitive to small differences in initial conditions. This is the butterfly effect that one may have heard about and it goes as follows: a small breeze generated from a butterfly’s wings in China may eventually mean a difference between a sunny day and a torrential rainstorm in California. This could be understood if one observed a chaotic system in phase space, because each unstable periodic motion would be shown to have corresponding trajectories in the phase space. Since there are a large number of these unstable periodic orbits in a bounded space, the trajectories corresponding to each of the unstable periodic orbits must be packed very closely together. In such close proximity, only a small change in the system is needed to push it from one orbit to any of a large number of others. This sensitivity of the system, jumping from one orbital trajectory to the next with slight perturbation input into the system or small differences in the initial conditions, has presented both a problem and a hope for those who deal with chaotic systems. This is a problem because, in the real physical system, knowing the exact values for all of the variables is impossible due to measuring devices’ limitations and inherent system noise. The consequence is that long term prediction of the system is unreliable or even impossible with any certainty. A good example of this can be found in weather predictions. For instance, one evening, the local weatherman predicts that the next day will be a beautiful sunny day. Only later in the afternoon, you find yourself walking in a thunder shower and not even having an umbrella to shelter yourself from this unexpected rain. The very property that cripples the long term prediction of a system can also be turned into a hope for dealing with the chaotic system. It becomes a hope because, if the system is so sensitive to even small changes, then small changes can cause the system to be controlled into a desired behavior such as periodic motion. This is what led Ott, Grebogi, and Yorke to formulate the chaos control scheme that has been proven to be successful in directing chaos to behave in any desired periodic motion.